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I have claim data structured by age groups and I am trying to test the assumed claim means for each age group against the actual data.

It is a mixture distribution where around 75% of the sample population has not experienced any claims and the rest ($X|X$>0) is highly skewed.

I am having a hard time deciding myself for a method and how to soundly justify my election. I have thought of the following:

1) Since if I sample my data, the distributions of the sample means look normal (and a saphiro test confirm they are) for most groups, I thought at first that a one-sample t-test was not a bad idea. Then again I read contradicting information on this (see below), and not being myself an statistician I gave up.

t-test on highly skewed data

Independent samples t-test: Do data really need to be normally distributed for large sample sizes?

(I realize their problems are different since I am not trying to compare the means of two samples).

2) Then I looked into bootstrapped CIs for the mean, first perfoming a calculation to determine the appropiate size of the samples for each group separately. The problem is I am unsure about bootstraping a realization of a subset of a population, specially given that some groups do not have much data (then again, in this case 1) is just as bad).

What do you think? How would you proceed, and how would you justify it?

Does it even make sense to try to validate such assumptions in this way?

I think it would make my life much easier (and it would be also more reasonable) if calculations were done on the basis of claim frequency ($E(X>0)$) and severity ($E(X|X>0)$), instead of claim mean $E(X)$, don't you think? Sadly it is not done this way and I cannot "separate" the assumption with the data I have.

A last question, even if I have a mathematics degree, I do not rely on myself when the statistics get a bit complex. Which resource could you recommend me to clear up these concepts, when data gets a bit tricky? I think an entry level book on inference would not help me (I have been already exposed to those).

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    $\begingroup$ "and a saphiro test confirm they are" - not a correct understanding of the result of the test -- failure to reject normality does not imply normality. The correct name of the test is Shapiro-Wilk. $\endgroup$
    – Glen_b
    Jun 13 '18 at 7:01

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